Output feedback sliding mode control based on adaptive sliding mode disturbance observer

In this paper, an adaptive sliding mode disturbance observer is designed to counteract the disturbance actively. By designing the adaptive laws, the assumptions on the disturbance are relaxed in the proposed observer, its first derivative upper bound is considered to be unknown. Based on the proposed disturbance observer, an output feedback sliding mode controller is constructed for the continuous-time linear systems with unknown external disturbance. The proposed controller incorporates only the system output information and has less chattering of the control input. The feasibility of the proposed strategy is shown by numerical simulations.

Sliding mode control (SMC) has attracted the attention of many researchers because of its robustness to disturbances and design simplicity. [1][2][3][4] In the area of SMC systems, many of the theoretical developments assume that the system state vector is available. [5][6][7] In most practical situations, it is unrealistic or prohibitively expensive to measure the full state information, and only the output information can be physically measured. [8][9][10] Generally, there are static and dynamic output feedback SMC strategies to work around this limitation. 11 The static output feedback SMC design problem is studied for a delay system. 12 Song et al. 13 address the static output-feedback sliding mode control (SMC) problem for a class of uncertain control systems. In order to improve the control performance of static output feedback SMC, the dynamic output feedback SMC can be designed with a compensator. The additional integral term is introduced in the sliding surface, which can provide one more degree of freedom. 14 An asynchronous output feedback sliding controller is proposed for a class of Markovian jump systems. 15 In the aforementioned results for output feedback SMC, the disturbance rejection problem is ignored, which leads to the chattering phenomenon in the system.
Recently, the observer-based method is utilized to eliminate disturbance and reduce chattering. 16, 17 Su et al. 18 present a disturbance observer where the bounds on disturbance derivative are assumed to be known. A new form of the combined observer-controller is designed to provide estimated data of unknown disturbance and unmeasured states in the control law. 19 Lee 20 proposed a composite control technique by combining a nonlinear disturbance observer. A fixed-time observer has been put forward, which possessed a better approximation to external disturbances. 21 However, in the above works the disturbance observer was considered with the assumption of the known maximum upper bound of disturbance. In some practical applications, it is difficult to acquire the disturbance upper bound directly, especially combined with the sliding mode algorithm, too large upper bound will aggravate the chattering phenomenon of the system, and too small upper bound will lead to the instability of the system. 1 The sliding mode disturbance observer with adaptive control is proposed to relax the restriction on disturbance. The adaptive control is introduced in the sliding mode disturbance observer that use the adaptive method to ensure the control gain is as small as possible whereas sufficient to eliminate the disturbances or uncertainties. 22 Negrete-Cha´vez and Moreno 23 develop an adaptive second-order SMC to interference disturbances. An adaptive scheme based on equivalent control is presented and it requires the disturbances upperbound is known. 24 Adaptive SMC methodology is structured which does not require a priori bounded uncertainty. 25,26 In order to further improve the observation performance of the observer, finite-time control methods were studied. 27 Finite-time convergence of the disturbance approximation error is guaranteed using the designed disturbance observer. 28 Compared with asymptotic convergence, 29 finite-time convergence can track the true value of the disturbance faster. The following three problems are interesting but challenging in this paper, (1) How to reduce the chattering of the output feedback controller, (2) how to design the disturbance observers with adaptive control to relax the constraints on disturbances, and (3) how to achieve finite-time convergence in the proposed observer.
The above-mentioned issue motivated us to devise a new output feedback sliding mode control based on the adaptive sliding mode disturbance observer strategy. Initially, the dynamic sliding surface for parameters analytically is designed, and an additional integral term is introduced in the sliding surface, which can provide one more degree of freedom. Based on the dynamic sliding mode surface, an output feedback controller is organized with power terms. It is worth mentioning that, the controller only contains the output information of the system and improves the convergence speed of the system state. In order to attenuate chattering in the control input, a novel ASMDO is developed. The proposed ASMDO does not require information about the bound on the disturbances and their derivatives. Compared to some existing observers, the conditions on disturbance bounds are relaxed. The effectiveness of the proposed control approaches is illustrated by an inverted pendulum on a cart model.

Preliminaries
Consider the following linear system with matched external disturbance where x(t) 2 R n is the state vector, u(t) 2 R m is the control input, y(t) 2 R q is the output vector, and m4q \ n. d(t) is the unknown disturbance. A 2 R n3n , B 2 R n3m , C 2 R q3n are given constant matrices.
When matched uncertainty alone is present, it is sufficient to consider the nominal linear system representation when designing the switching function. The sliding motion depends on the choice of the sliding surface, the precise effect is not readily apparent. Therefore, we need to transform the system into a suitable regular form.

Design of dynamic sliding surface
The dynamic sliding surface is designed as An additional integral term n(t) is introduced in the sliding surface, which can provide one more degree of freedom. n(t) is defined as follows and w 1 2 R m3(qÀm) , w 2 2 R (nÀm)3(qÀm) , w 3 2 R (nÀm)3(nÀm) , w 4 2 R m3(nÀm) are the sliding surface parameter matrices, F 2 R (nÀm)3m is a known matrix. The derivative with respect to (8) then (7) can be rewritten as According to (10), we have Substituting (11) into (4) yields the sliding motion equation Define W 2 = w 2 + Fw 1 and W 3 = w 3 À Fw 4 , (9) can be rewritten into the following form Thus, the new sliding motion is as follows where ! and , H 1 and H 2 will be introduced later. The new sliding motion is asymptotically stable on the dynamic sliding surface, if there exist F 1 2 R (nÀm)3(nÀm) , G 1 2 R (nÀm)3(nÀm) are symmetric positive definite matrices and subject to Proof. Define G 2 and F 2 are nonsingular matrices, and satisfies G 2 F 2 = I À G 1 F 1 : Coordinate transformation matrices H 1 and H 2 are designed as follows hence exist a positive definite symmetric matrix F . 0 such that The new sliding motion (14) is asymptotically stable. Ä

Design of sliding mode controller
In this subsection, the control method based on output feedback sliding mode with power term is proposed.
Define the generalized inverse matrix of C 1 is C À 1 and z 1 = null(C 1 ) T satisfying that z 1 C T 1 = 0, generalized inverse matrix of z 1 is z À 1 , then equation C À 1 C 1 + z À 1 z 1 = I nÀm holds. Since A is Hurwitz matrix, then exist j . 0 and P . 0 subject to Denote j as the largest positive scalar such that (17) is feasible. Utilize Cholesky factorization for P to get Remark 1. Inequality (17) holds which can be explained as follows: Consider the Lyapunov function for the new sliding motion (14): Since A is Hurwitz matrix and P . 0, j . 0, hence _ V \ 0, PA + A T P \ 0. There exists j . 0 for (PA + A T P) + 2jP \ 0 holds.
0, then the following output feedback sliding mode controller is proposed: where c 1 (t) = c 0 (q 1 (t) + .) and d is the estimation of the disturbance. Then, the system (2)2(3) can reach the sliding surface.
Proof. The first derivative of the sliding mode surface is given as In order to prove that s(t) can converge to 0, define the Lyapunov function V s (t) = 1 2 s T (t)s(t), its derivative can be determined and substitute (19) Substituting (18) into (20), one obtains where x(t) = c 0 kP T 11 x 1 (t) + P T 21 n(t)k À(k 1 + k 2 )c 1 (t): Obviously, Àt ks(t)k 2 \ 0, and assumed À d(t) = 0 holds. In order to ensure that _ V s (t) \ 0 holds, we need to discuss the positive and negative of x(t).
This implies that ifd À d(t) = 0 is guaranteed, then s(t) will converge to zero and the system can reach the sliding surface. where c 1 , c 2 are positive constants, 0 \ a 1 \ 1 and a 2 . 1. The equilibrium point of the above system is finite time stable and its settling time is bounded.
Design sliding mode disturbance observer In this section, the sliding mode disturbance observer is designed to guarantee s(t) converge to zero. The following auxiliary variable is introduced The coefficients c 1d , c 2d , a 1d , a 2d are positive constants with a 1d \ 1, a 2d . 1. y z will be introduced later. If y z satisfies the following equation where h d is a positive constant. k d (t) . d 1 , d 1 . jB 2 _ dj and d 0 . jB 2 dj. d and _ d are disturbance and its derivatives, d 1 and d 0 are the upper bounds of jB 2 _ dj and jB 2 dj , respectively. e o converges to origin in finite time and the estimation of the disturbance as follows Proof. Consider the derivative of e o in equation (32) as follows Substitute (37) into (34) one obtains Differentiating the above equation (38) and combining with equation (35) yields Consider the Lyapunov function V d (t) = 1 2 s 2 d (t), then V d (t) derivative is as follows Since h d is a positive constant and k d (t) . d 1 . jB 2 _ d(t)j, then (40) can be written as follows The s d (t) will converges to zero in finite time, by utilizing equation (34) one gets _ e o (t) = À c 1d sgn(e o (t))je o (t)j a 1d À c 2d sgn(e o (t))je o (t)j a 2d By considering Lemma 1, the equilibrium of (42) as e o (t) = 0 is finite time stable.
Consider the estimation error of disturbance as follows Both (36) and (38) are satisfied, e d (t) will convergence to origin in finite time. This implies that if k d (t) . d 1 is satisfied, the proposed disturbance observer can estimate the disturbance d(t) in finite time.

Design adaptation structure
In this subsection, two adaptation structures are designed, and the assumptions on the disturbance are relaxed in the proposed observer. As is capable of seeing from the proof in the above subsection, the adaptive gain k d (t) must satisfy k d (t) . d 1 to ensure the system sliding takes place, in which case reachability condition is achieved 30 and sliding motion takes place on s d = _ s d = 0. In other words, unknown disturbance or uncertainty should be completely eliminated, that is ju eq (t)j = jd 1 j: In the first adaptive structure, similar to general adaptive control, 24 it is assumed that d 1 is known and d 0 is unknown. The second adaptive strategy is designed which assumes that both d 1 and d 0 are unknown, the assumptions on the disturbance are more relaxed.
Consider utilizing a low pass filter to filter the switching signal to obtain a close approximation. Here, the sgn(s d ) should take on the average value. Then if u eq (t) satisfies where l . 0 is a time constant, thenû eq (t) almost completely approximates u eq (t). In order to eliminate the influence of the initial conditions of the filter, assume exist 0 \ y 1 \ 1 and y 0 . 0 such that jjû eq (t)j À ju eq (t)jj \ y 1 ju eq (t)j + y 0 holds. The adaptive algorithm of the control gain is driven by using the equivalent control. Through the following inequality, we introduce the concept of 'safety margin' where 0 \ m \ 1 and y . 0 are design parameters such that 1 m jû eq (t)j + y 2 . ju eq (t)j ð 47Þ holds. Define error variable e k (t) as follows which shows that if e k (t) = 0, then k d (t) . d 1 , that is, the system will maintain sliding motion. Define adaptive scheme where g(t) is auxiliary scalar, it has the following form where f 0 is a positive scalar and f(t) is elaborated later. In this paper, according to whether d 1 is known, f(t) will execute different choices. In the next work, we will discuss two situations: d 1 is known and d 1 is unknown.
where b . 0, and a . 1 is designed to ensure j _ u eq (t)j \ ad 1 . Proposition 4. As described above, the problem of maintaining sliding is converted to how to yield e k (t) ! 0 in finite time; how to allow g(t) and k d (t) to be bounded. In the following, we will verify the situation when d 1 is known.
Proof. Consider following Lyapunov function It follows from (53) that The derivative with respect to (53) and integrate with (54) and (55), one obtains Define V 0 to be the initial value of V(t) and t 0 to be the time taken for V(t) converge to zero, and hence integrating both sides yields t 0 \ ffiffiffiffiffiffi 2V 0 p f 0 , which shows that e k (t) and e r (t) will converge to origin in finite time. Then k d (t) . d 1 will be guaranteed and the reachability condition is satisfied. Obviously e r (t) is bounded, hence f(t) and g(t) is bounded.
Case 2: Assuming that both d 1 and d 0 are unknown. Define where e 0 is a design constant which to eliminate the noise signal in the system.
then je k (t)j \ y 2 is realized in finite time and the sliding motion is guaranteed.
Proof. Consider the Lyapunov function from (53) e k _ e k 4 À f 0 je k (t)j + e r (t)je k (t)j ð 59Þ Suppose f(0) = 0, from (57) one obtains _ f(t)50, then f(t)50 holds. Since e r (t) = ad 1 m À f(t), then e r (t)4 ad 1 m is satisfied. In (57), if je k (t)j . e 0 with (55), _ e r (t) = À _ f(t) = À bje k (t)j is established, (56) can be rewritten as If je k (t)j \ e 0 with e r (t) \ 0, then _ V(t)4 À : Choose the appropriate y to satisfy (58), the sliding motion will is guaranteed. Since E & F and outside of F in the solution domain, _ V(t)40 holds, hence F is an invariant set. If the V(e k , e r ) enters F, then V(e k , e r ) will not be able to leave F and from (58), je k (t)j \ y 2 is satisfied. If the V(e k , e r ) does not enter F, then from the above discussion _ V4 À f 0 je k (t)j and Ð ' 0 f 0 je k (t)jdt4V(0) are satisfied. Since V(e k , e r ) is bounded, then e k (t), e r (t) and _ e k (t), _ e r (t) are bounded, hence je k (t)j is uniformly continuous, that is, e k (t) ! 0 when t ! '. It shows that there exists t 0 such that je k (t)j 4 y 2 for t . t 0 . Hence, je k (t)j \ y 2 has always been established in finite time. Since the je k (t)j \ y 2 holds, therefore je k (t)j = jk d (t) À 1 m jû eq (t)j À yj \ y 2 from (48), that is k d (t) À 1 m jû eq (t)j À y . À y 2 , and from (47), one obtains Hence reachability condition is satisfied and the system will maintain sliding motion. Since e k (t) and e r (t) are bounded, consequently f(t) and g(t) remains bounded. From (45) obviously k d (t) is bounded, and the proof is complete.
Remark 2. The method proposed in this paper can be improved by some new fuzzy systems, at the same time, the ASMDO does not require knowledge of the disturbance and its first derivative, hence it can better estimate the disturbances in these systems. 31,32 Remark 3. The control strategy of this paper can be used to deal with the problem of energy/voltage management in photovoltaic (PV)/battery systems 33 and interval type-3 fuzzy logic systems. 34 The sliding mode disturbance observer can be used to eliminate the influence of the variation of temperature, radiation, and output load on the system, and effectively guarantee better disturbance rejection performance of the controller.

Simulation example
A practical example is provided in this section to illustrate the efficiency of the obtained result. Consider an inverted pendulum on a cart model given in Edwards and Spurgeon 8 and the linearization model is as follows The non-singular matrix T can be found as The initial condition is x(0) = ½0:1 0:01 0:2 0:15. This system is simulated with disturbance: d(t) = 2sin(2t) + 2cos(2t). The parameters of the sliding mode disturbance observer are selected as c 1d = c 2d = 2, a 1d = 0:5, a 2d = 3, h d = 0:01. The disturbance is set to B 2 _ d(t) = j2sin(pt) + 0:15cos(2pt)j. The parameters of adaptive schemes are chosen as d 1 = 1, m = 0:99, f 0 = 0:6, e 0 = 0:01, y = 0:18, l = 0:01, and starting with zero initial conditions. Figures 1 and 2 are the control input of the system obtained from Zhang et al. 11 and (18), respectively. Obviously, the proposed controller possesses less chattering. The fluctuation of the control input from 0 to 6 s is caused by the system states x 1 (t) and x 2 (t) in the controller (18). It can be seen from Figure 4 that when the system state tends to be stable, the input of the controller will also tend to stable. The fluctuation of control input after 6 s is caused by k 1 sgn h1 (s(t)) + k 2 sgn h2 (s(t)) in the controller (18). Figure 3 shows that after the sliding surface converges to 0, there is a slight fluctuation around zero, which causes the sgn(s) in the controller to switch between 21 and 1, thus generating the fluctuations in Figure 2. Figure 4 depicts that system state by Zhang et al. 11 and (18), respectively. Obviously, the system states obtained with the proposed controller have less overshoot and faster convergence performance.  The coefficient of additional term k 1 + k 2 . 1 located in denominator, although the item is omitted in t s , actually, it must enable the system to converge faster. Figure 5 is evolutions of disturbance and its estimation in controller. Compared with Hwang et al., 29 the estimation of disturbance can faster track its true value in finite time under the proposed observer (33). Figures  6 and 7 shown that the adaptive gain k d (t) still closely follows B 2 _ d(t). As in (48), k d (t) converges to a safety margin which depends on the parameters m and y. k d (t) is always above B 2 _ d(t), the sliding motion will be maintained and the conditions on disturbance bounds are relaxed, the proposed ASMDO does not require information about the bound on the disturbances and their derivatives.

Conclusion
This paper has addressed a disturbance observer-based control method for continuous-time linear systems with unknown external disturbance. Based on a novel adaptive sliding mode disturbance observer, the output feedback sliding mode controller has been designed, which guarantees that reachability condition holds strictly. In the absence of upper bound information of the disturbance and its first derivative, the restrictive restraints on disturbance have been relaxed by designing the adaptive laws. It is worth mentioning that the estimation of disturbance can track its true value in finite time under the proposed observer. An inverted pendulum on a cart model has been exploited to illustrate the effectiveness of the proposed controller and observer. Simulation results show that the convergence performance of the controller can be further improved, and the finite time control observer-based method will be studied in future.